Randomized Algorithms and Lower Bounds for Quantum Simulation

We consider deterministic and {\em randomized} quantum algorithms simulating $e^{-iHt}$ by a product of unitary operators $e^{-i{A}_j{t}_j}$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},N$, where $A_j\in \{H_1,\mathrm{.}\mathrm{.}\mathrm{.},H_m\}$, $H={\sum }_{i=1}^m{H}_i$ and $t_j>0$ for every $j$. Randomized algorithms are algorithms approximating the final state of the system by a mixed quantum state. First, we provide a scheme to bound the trace distance of the final quantum states of randomized algorithms. Then, we show some randomized algorithms, which have the same efficiency as certain deterministic algorithms, but are less complicated than their opponentes. Moreover, we prove that both deterministic and randomized algorithms simulating $e^{-iHt}$ with error $\text{e}$ at least have $\mathrm{\Omega }(t^{3\mathrm{/}2}{\text{e}}^{-1\mathrm{/}2})$ exponentials.